The QOC pulses were optimized using a Python implementation of the GRAPE algorithm with numerical efficiency boosted by efficient spin control using analytical Lie algebraic derivatives (ESCALADE) . The SciPy implementation of the limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm (L-BFGS-B) was chosen as optimization back-end . To facilitate selectivity with respect to ν0 and ν1, we used the concept of pattern pulses, a special variant of ensemble QOC where pulse optimization runs simultaneously across a whole ensemble of spin systems, each corresponding to a different combination of effective ν0 and ν1 . By assigning a target spin state ρtarget, corresponding to excitation or suppression, to each ensemble element, excitation patterns are achieved. Each ensemble element yields a quality factor computed as Re〈ρtarget|ρfinal〉, the real part of the scalar product between the final spin state ρfinal after applying the pulse and ρtarget. The total quality function optimized by L-BFGS-B was computed as the weighted average of the ensemble quality factors. If not specified otherwise, the ensemble elements were weighted equally. The total control amplitude was limited by a hard upper bound of 10 kHz. Robustness with respect to B0 variations was incorporated by variation of the Larmor frequencies ν0, whereas robustness with respect to B1 inhomogeneities was incorporated by allowing linear scalings of the nutation frequencies ν1. The arising 2D grid of varying parameters is termed the excitation profile. The excitation profile resolution was set to 41 ν1 scalings times 401 ν0 offsets. The respective linear relation between B0 and ν0, as well as between B1 and ν1 , is given by 1ν=-γB2π. In this work, B0 and B1 in the vicinity of different materials are determined by FEM simulations. The B and ν terms are used interchangeably in the following unless an unambiguous distinction is required and stated. It must be noted that, instead of B0, the effective magnetic field that acts on a nuclear spin is used in this paper. That includes chemical (de-)shielding effects affecting the nucleus, as well as susceptibility effects of any material in its environment. Thus, B0, as used in this paper, may differ from the external, static magnetic field.

For the optimization of a ν0-selective, ν1-robust excitation pulse, the excitation profile was chosen such that spins within a ν0 range between -500 and 500 Hz are excited with ρtarget=Ix, whereas the remaining spins within the ν0 range from -2 to 2 kHz are suppressed, corresponding to ρtarget=Iz. Offsets outside of the selected bandwidth were not controlled. For a ν0-selective suppression pulse, the target states were swapped. The linear scaling factors of ν1 ranged between 0.9 and 1.6. For the excitation pulse, a duration of 1 ms with 2000 equidistant time increments of 0.5 µs was sufficient to achieve a final mean quality factor of 94.9 %. For the suppression pulses, two distinct parameter sets were utilized: a duration of 1 ms with 2000 equidistant time increments of 0.5 µs (mean quality factor of 93.5 %) and a duration of 2 ms with 4000 equidistant time increments of 0.5 µs (mean quality factor of 95.6 %) to achieve a better frequency selectivity.

In the case of the suppression pulse, the quality factors of ensemble elements corresponding to a ν0 between -250 and 250 Hz were weighted by a factor of 5 relative to the remaining fidelities. Furthermore, the quality factors of ensemble elements corresponding to a ν1 scaling between 0.9 and 1.1 were additionally weighted by a factor of 5. This was due to the fact that the ensemble elements associated with these regions of the excitation profile tended to reach insufficient quality factor values for a homogeneous signal suppression. The pulse shapes and excitation profiles of the ν0-selective excitation and suppression pulses are visualized in Sects. S1.1 and S2.1 in the Supplement (Figs. S1 to S3 and S10 to S12), respectively.

The ν1-selective, ν0-robust pattern pulse excitation profiles were set up based on the increase in B1 intensity around metallic elements. The increase was quantified by FEM simulations of the local B1 field in the copper and polymer (represented by vacuum) cavities of the model setup following the procedure described in . The simulated geometry is visualized in Sect. S3 (Fig. S19). The predicted relative B1 increase is then determined as the ratio 2ΓB1=B1(Cu)B1(vacuum). For a given ΓB1, the excitation region in the profile is defined as the set of ensemble elements corresponding to ΓB1±0.1. Ensemble elements in the excitation region are assigned ρtarget=Ix, and the remaining elements are assigned ρtarget=Iz. The ν0 offset range was chosen to be from -1 to 1 kHz. The range of ν1 scalings was chosen to be from 0.9 to 1.6, except for the pulse for selective excitation at ΓB1=1.8, where the maximum ν1 scaling was raised to 1.9. A pulse duration of 1 ms with 2000 equidistant time increments of 0.5 µs was sufficient for all ΓB1. The obtained final quality factors are listed in Table . The pulse shapes and excitation profiles of the ν1-selective excitation and suppression pulses are visualized in Sects. S1.2 and S2.2 (Figs. S4 to S9 and S13 to S18), respectively.

A model setup was prepared to demonstrate the operating principle of the developed QOC pulses. To investigate the field-distorting effects of conductive materials on liquid samples, the setup contained two cavities of equal dimensions: the first one in between two copper coins and the second one in between two polymer coins made of polyether ether ketone (PEEK), serving as the reference cavity with minor magnetic field distortions. The rotation axes of both cylindrical cavities and their coins were oriented parallel to the B0 field and perpendicular to the B1 field, in accordance with the findings in a previous publication . The model setup was fitted into a shortened common 5 mm NMR tube and is illustrated in Fig. a and b, including the diameter and thickness of copper and polymer coins, as well as the polymer spacers which defined the size of the cavity. The NMR tube was shortened to remove the narrowed opening such that the entire tube has a diameter of 4 mm.

Liquids were only added in between two coins of equal material, limiting the maximum filling height of liquid in each cavity to 0.1 mm. n-dodecane (in copper cavity) and H₂O (in PEEK cavity) were chosen as non-mixable liquids, each of them filled into one of the two cavities, to enable an unambiguous distinction by spatial position, as well as by their chemical shift. The model setup was positioned in the NMR tube such that the two cavities were placed around the center of the NMR-sensitive volume.

A Bruker DiffBB BBO broadband diffusion probe and a Bruker Diff50 ¹H diffusion probe, both with magnetic field gradients along the B0 direction (z axis) with a maximum gradient strength of 2312 Gcm-1 were operated on a Bruker Avance III HD spectrometer (Bruker BioSpin GmbH, Rheinstetten, Germany) with a 9.4 T wide-bore magnet.

The spatial position of the two liquids was verified by phase-encoded ¹H chemical shift imaging (CSI) along the z axis (Fig. b). Phase-encoded ¹H CSI was performed by using pulsed magnetic field gradients along z with a strength of ±37.62 Gcm-1. The field of view was set to 20 mm with 32 points in the spatial dimension, resulting in a spatial resolution of 625 µm. Due to sharp susceptibility changes at interfaces, homogeneous shimming over the sample volume was not possible. Instead, the shims were adjusted for more pronounced separation of the resonances of the two compounds. No chemical shift reference was integrated into the model setup. Thus, the chemical shifts of water and n-dodecane do not correspond to their tabulated values.

Standard FID (free induction decay)-detected NMR experiments using a single hard pulse adjusted for a Bruker reference sample were compared to experiments where the hard pulse was substituted by QOC pulses. For the ν0-selective QOC pulses, in order to illustrate the selective bandwidth of 1000 Hz over the total pulse bandwidth of 4000 Hz, the ¹H resonance frequency offset Δν0 was varied in steps of either 200 Hz or 300 Hz, starting 600 or 900 Hz downfield of the H₂O resonance for a total frequency range of 2200 Hz or 2700 Hz, respectively. For the ν1-selective pulses, Δν0 was set such that the total pulse bandwidth of 2000 Hz contained the resonances of both n-dodecane and H₂O. However, it was found that the pulse length required for a 90° pulse in the presented model setup differs from a Bruker reference sample. The actual 90° pulse lengths for each cavity were determined by a nutation experiment, and the resonance integrals in all reference spectra were adjusted accordingly. The acquisition time was adjusted to record the full FID, and the recovery delay was set to allow for complete relaxation. The baseline was corrected by using a splines fit.

Nutation experiments were performed with rectangular pulses of varying pulse length and a constant pulse power of 1.5 W for the Diff50 probe and 3.8 W for the DiffBB probe. In total, 100 pulse lengths were screened, with a step size of 30 µs.

The ν0-selective QOC excitation pulses with a pulse length of 1 ms were demonstrated in a proof-of-concept experiment on the described model setup, with n-dodecane in the copper cavity and H₂O in the PEEK cavity. Figure  illustrates the application of the ν0-selective pulses at selected offsets Δν0. Here, Δν0 was varied with a step size of 200 Hz between 2975 and 775 Hz (marked with a green dot and arrow), where the excitation is centered on either the resonance of H₂O for Δν0=2375 Hz and on the resonance of n-dodecane for Δν0=1375 Hz. The top spectrum was recorded utilizing a hard pulse and serves as a reference for comparison. The linewidth of the H₂O resonance (50 Hz) is smaller compared to the coalesced n-dodecane resonance (150 Hz). Both resonances show broad features due to a non-optimal shim caused by the significant magnetic susceptibility gradients throughout the model setup. Thus, the error on each individual resonance integral was calculated from the respective signal-to-noise ratio of each experiment.

When centering the excitation band on either resonance, the QOC pulse achieved efficient excitation, yielding a relative integral of 87.44±0.43 % for H₂O or 115.34±0.26 % for n-dodecane compared to the 90° hard pulse. Simultaneously, the respective other resonance was suppressed to either 4.96±0.28 % or 4.02±0.40 % of its original value. The relative QOC integrals of both resonances compared to the reference for each of the ν0-selective pulses are summarized in Table S1 in the Supplement.

Analogously, ν0-selective QOC suppression pulses with the same pulse length were applied with a Δν0 step size of 300 Hz. The pulses (Fig. S27 and Table S2) efficiently suppressed the H₂O resonance to 7.42±0.97 % of its original value. However, the n-dodecane resonance was only suppressed to 32.66±0.32 %. This was due to an insufficient pulse performance at a duration of 1 ms, a non-optimal Δν0 (+100 Hz) for n-dodecane in this specific experiment, and the large n-dodecane linewidth. To increase the suppression efficiency, the pulse duration was extended to 2 ms, and the step size was readjusted to 200 Hz such that the suppression band was centered on n-dodecane or H₂O. In the adjusted experiment (Fig.  and Table S3), H₂O was suppressed to 1.01±0.48 % and n-dodecane was suppressed to 1.29±0.26 % of their respective reference value when centering the suppression band on either of them, yielding a significantly improved suppression.

To assess the signal selectivity of the QOC excitation pulses quantitatively, a selectivity parameter 3SQOCexc=Iexc(Isup,ref-Isup)Iexc,refIsup,ref was defined, where Iexc and Isup denote the integrals of the excited and suppressed resonances obtained by the QOC pulse, respectively. Analogously, Iexc,ref and Isup,ref denote the integrals of the excited and suppressed resonances obtained by the hard 90° pulse, respectively. SQOC is between 0 and 1 in the ideal case where the 90° hard-pulse reference achieves a uniform maximum excitation, where SQOC=1 describes an optimally selective and SQOC=0 a non-selective QOC excitation. However, QOC excitation surpassing the excitation of 90° hard pulses was observed experimentally, thus achieving SQOC>1 in some cases. We believe the origin of this effect lies within the capacitive interaction of the rf pulse B1 and the conductive plates, which is more pronounced for the short, hard pulse than for the long, soft QOC pulse.

A corresponding selectivity parameter 4SQOCsup=Isup(Iexc,ref-Iexc)Iexc,refIsup,ref was defined for QOC suppression pulses. In the following, SQOC will be used to describe the selectivity of all QOC pulses and excitation, as well as suppression, referring to their respective SQOCexc or SQOCsup.

The highest SQOC for ν0-selective excitation in Fig.  amounted to 1.202±0.007 when positioning Δν0 200 Hz upfield from the center of the n-dodecane resonance (Δν0=1175 Hz, 11th spectrum from top). This was slightly higher compared to SQOC=1.107±0.006 when the pulse was exactly on-resonance (Δν0=1375 Hz, 10th spectrum from top). Comparatively, selective suppression in Fig.  achieved a maximum SQOC of 1.177±0.008 with Δν0 200 Hz upfield from the center of the n-dodecane resonance (Δν0=975 Hz, 12th spectrum from top), also slightly higher compared to SQOC=1.096±0.007 when exactly on-resonance (Δν0=1175 Hz, 11th spectrum from top). Herein, excitation pulses achieved a similar maximum SQOC as suppression pulses.

The achievable SQOC in Fig.  was strongly affected by Δν0, even when within the pulse robustness range of ±500 Hz. For example, SQOC is reduced to 0.397±0.004 and 0.839±0.006 when moving Δν0 either downfield (Δν0=1775 Hz, 8th spectrum from top) or upfield (Δν0=975 Hz, 12th spectrum from top) by 400 Hz, respectively. SQOC is also limited by the proximity of the suppressed and excited frequency ranges. The closer the resonance frequencies are to each other, the more challenging a selective excitation or suppression via ν0-selective QOC pulses becomes, similarly to the case for conventional selective pulses . A near-instant transition from excitation to suppression along the ν0 dimension is difficult to realize in the pulse optimization and requires longer pulse durations for sharper transitions.

To overcome the selectivity difficulties for small frequency ranges, B1 field variances originating from conductive materials were exploited instead of ν0 differences in the optimization of QOC pulses.

The FEM-predicted B1 increase was validated experimentally via a nutation experiment (Fig. S20). A comparison of the obtained nutation frequencies ν1 for both cavities in the model setup is shown in Fig. . The FEM simulations revealed two narrow, clearly separated B1 distributions for the copper and PEEK cavity with ΓB1FEM=1.257. The experimentally determined ΓB1exp amounted to 1.256, resulting in a difference of 0.001 between simulations and experiments. A difference of this magnitude is negligible when applying QOC pulses due to their robustness with respect to B1 inhomogeneities of ±0.1. Exchanging H₂O by n-dodecane or other liquids affected neither the simulated nor the experimentally determined ΓB1 (Sect. S4.2.1). Thus, the difference in B1 clearly originates from the surrounding material.

To exemplify the accuracy of the method, a range of QOC pulses for varying ΓB1 with ν0 robustness (Sect. ) was applied to the model setup and compared for a systematical screening of pulse effectiveness (Fig. ). The errors of all values were calculated from the respective signal-to-noise ratio of each experiment.

A 90° hard pulse reference spectrum of the same setup was recorded for comparison. The relative excitations of both resonances compared to the reference for each of the ν1-selective pulses are summarized in Table .

The QOC pulses for ΓB1=1.2, ΓB1=1.25 and ΓB1=1.3 selectively excited the n-dodecane resonance to a minimum of 90.06±0.49 % and simultaneously suppressed the resonance of H₂O to a maximum of 11.57±0.60 %. These results confirm a successful QOC excitation of resonances inside the copper cavity while suppressing resonances inside the PEEK cavity within the ΓB1 robustness limit of ±0.1. The highest excitation of n-dodecane was achieved with the pulse for ΓB1=1.25, while the highest suppression of H₂O was achieved with the pulse for ΓB1=1.30.

In comparison, the application of a QOC pulse optimized for ΓB1=1.0 excited the H₂O resonance to 99.79±0.49 % and suppressed the n-dodecane resonance to 16.53±0.30 % of their original values. This illustrates that, depending on the ΓB1 which the QOC pulses are optimized for, a resonance selectivity for either the copper cavity with increased B1 or the PEEK cavity with no B1 increase can be achieved. Furthermore, the additional spectra corresponding to even higher ΓB1 exemplify how a mismatch between the ΓB1 range of the QOC pulse and the B1 experienced by both investigated spin systems leads to both resonances being suppressed to a maximum of 14.09±0.33 % (ΓB1=1.4) and 3.63±0.34 % (ΓB1=1.8) of their original values.

SQOC was again determined to evaluate the effective signal selectivity of the ν1-selective QOC pulses and to compare it to the ν0-selective QOC pulses. Additionally, the experiments shown in Fig.  (except ΓB1=1.25) were each repeated three times for the determination of a standard deviation. Thus, the standard deviation will be used instead of the error based on the signal-to-noise ratio to evaluate the QOC pulse performance in the following. The average SQOC values are summarized in Table .

The highest average SQOC was found for ΓB1=1.2 at 1.07±0.23. For ΓB1=1.3, the signal selectivity was, on average, slightly lower at SQOC=0.96±0.05. Comparing the theoretical ΓB1 of these pulses suggests that the pulse for ΓB1=1.3 should be more selective due to its ΓB1 being slightly closer to the experimentally determined value of 1.256. For the ΓB1=1.2 pulse, the experimental ΓB1=1.256 is therefore further away from the transition region of the QOC pulse, where excitation of the copper cavity transits to suppression of the PEEK cavity (Fig. S14) at 1.1 in contrast to the ΓB1=1.3 pulse with a transition at 1.2 (Fig. S16). However, the SQOC deviation between the pulses for ΓB1=1.2 and ΓB1=1.3 lies in the range of the error bars. By comparison, QOC pulses for ΓB1 outside of the expected range result in a significantly lower average SQOC of 0.166±0.040 (ΓB1=1.4) and 0.029±0.005 (ΓB1=1.8); thus, the selectivity parameter SQOC precisely and reproducibly expresses the QOC pulse selectivity.

For completeness sake, we also tested the impacts of different rf pulse center frequencies, B0 field shim, and receiver gain (Sect. S4.2.2 to S4.2.4) on the QOC spectra, which were revealed to be negligible, thus adding additional flexibility to the experimental implementation of QOC pulses.